document.write( "Question 681794: Help explain why this expression cannot be factored
\n" ); document.write( "40a^2+21a-2 \n" ); document.write( "

Algebra.Com's Answer #422798 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"40a%5E2%2B21a-2\"$ , we can see that the first coefficient is $\"40\"$ , the second coefficient is $\"21\"$ , and the last term is $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"40\"$ by the last term $\"-2\"$ to get $\"%2840%29%28-2%29=-80\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-80\"$ (the previous product) and add to the second coefficient $\"21\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-80\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-80\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,40,80\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-40,-80\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-80\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-80) = -80
\n" ); document.write( "2*(-40) = -80
\n" ); document.write( "4*(-20) = -80
\n" ); document.write( "5*(-16) = -80
\n" ); document.write( "8*(-10) = -80
\n" ); document.write( "(-1)*(80) = -80
\n" ); document.write( "(-2)*(40) = -80
\n" ); document.write( "(-4)*(20) = -80
\n" ); document.write( "(-5)*(16) = -80
\n" ); document.write( "(-8)*(10) = -80\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"21\"$ :\r
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First Number	Second Number	Sum
1	-80	1+(-80)=-79
2	-40	2+(-40)=-38
4	-20	4+(-20)=-16
5	-16	5+(-16)=-11
8	-10	8+(-10)=-2
-1	80	-1+80=79
-2	40	-2+40=38
-4	20	-4+20=16
-5	16	-5+16=11
-8	10	-8+10=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"21\"$ . So $\"40a%5E2%2B21a-2\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"40a%5E2%2B21a-2\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"40a%5E2%2B21a-2\"$ is prime.
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